Perturbative reheating and thermalization of pure Yang-Mills plasma

We investigate the thermalization of high-energy particles injected from the perturbative decay of inflaton during the pre-thermal phase of reheating in detail. In general, thermalization takes a relatively long time in a low-temperature plasma; therefore, the instantaneous thermalization approximation is not justified, even for the reheating of the Standard Model (SM) sector. We consider a pure Yang-Mills (YM) theory as an approximation of the SM sector or a possible dark sector, considering the Landau-Pomeranchuk-Migdal effect, a quantum interference effect in a finite temperature plasma. We perform the first numerical calculation to solve the time evolution of the system, including the redshift due to the expansion of the Universe, and show the details of the temperature evolution near the maximum and the behavior of the quasi-attractors at later times. The maximal temperature $T_\text{max}$ and time scale $t_\text{max}$ are determined quantitatively, such as $T_\text{max} \simeq 0.05 \times (\Gamma_I M_\text{Pl}^2/m_I^3)^{2/5} m_I$ and $t_\text{max} \simeq 2 \times 10^3 \times (\Gamma_I M_\text{Pl}^2/m_I^3)^{-3/5} m_I^{-1}$ in the SM-like system, where $m_I$ and $\Gamma_I$ are the mass and decay rate of inflaton. We also provide a similar formula for pure $\operatorname*{SU}(N)$ and $\operatorname*{SO}(N)$ YM theories for general values of $N$ and coupling constant $\alpha$, including $T_\text{max} \propto \alpha^{4/5}$ and $t_\text{max} \propto N^{-2} \alpha^{-16/5}$ behaviors and their numerical coefficients. The thermalization occurs in a finite time scale, resulting in a lower maximal temperature of the Universe after inflation than that under the instantaneous thermalization approximation.